text: /* Copyright (C) 1995-1998 Eric Young (eay@cryptsoft.com)
   * All rights reserved.
   *
   * This package is an SSL implementation written
   * by Eric Young (eay@cryptsoft.com).
   * The implementation was written so as to conform with Netscapes SSL.
   *
   * This library is free for commercial and non-commercial use as long as
   * the following conditions are aheared to.  The following conditions
   * apply to all code found in this distribution, be it the RC4, RSA,
   * lhash, DES, etc., code; not just the SSL code.  The SSL documentation
   * included with this distribution is covered by the same copyright terms
   * except that the holder is Tim Hudson (tjh@cryptsoft.com).
   *
   * Copyright remains Eric Young's, and as such any Copyright notices in
   * the code are not to be removed.
   * If this package is used in a product, Eric Young should be given attribution
   * as the author of the parts of the library used.
   * This can be in the form of a textual message at program startup or
   * in documentation (online or textual) provided with the package.
   *
   * Redistribution and use in source and binary forms, with or without
   * modification, are permitted provided that the following conditions
   * are met:
   * 1. Redistributions of source code must retain the copyright
   *    notice, this list of conditions and the following disclaimer.
   * 2. Redistributions in binary form must reproduce the above copyright
   *    notice, this list of conditions and the following disclaimer in the
   *    documentation and/or other materials provided with the distribution.
   * 3. All advertising materials mentioning features or use of this software
   *    must display the following acknowledgement:
   *    "This product includes cryptographic software written by
   *     Eric Young (eay@cryptsoft.com)"
   *    The word 'cryptographic' can be left out if the rouines from the library
   *    being used are not cryptographic related :-).
   * 4. If you include any Windows specific code (or a derivative thereof) from
   *    the apps directory (application code) you must include an acknowledgement:
   *    "This product includes software written by Tim Hudson (tjh@cryptsoft.com)"
   *
   * THIS SOFTWARE IS PROVIDED BY ERIC YOUNG ``AS IS'' AND
   * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
   * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
   * ARE DISCLAIMED.  IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
   * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
   * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
   * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
   * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
   * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
   * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
   * SUCH DAMAGE.
   *
   * The licence and distribution terms for any publically available version or
   * derivative of this code cannot be changed.  i.e. this code cannot simply be
   * copied and put under another distribution licence
   * [including the GNU Public Licence.]
   */
  /* ====================================================================
   * Copyright (c) 1998-2005 The OpenSSL Project.  All rights reserved.
   *
   * Redistribution and use in source and binary forms, with or without
   * modification, are permitted provided that the following conditions
   * are met:
   *
   * 1. Redistributions of source code must retain the above copyright
   *    notice, this list of conditions and the following disclaimer.
   *
   * 2. Redistributions in binary form must reproduce the above copyright
   *    notice, this list of conditions and the following disclaimer in
   *    the documentation and/or other materials provided with the
   *    distribution.
   *
   * 3. All advertising materials mentioning features or use of this
   *    software must display the following acknowledgment:
   *    "This product includes software developed by the OpenSSL Project
   *    for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
   *
   * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
   *    endorse or promote products derived from this software without
   *    prior written permission. For written permission, please contact
   *    openssl-core@openssl.org.
   *
   * 5. Products derived from this software may not be called "OpenSSL"
   *    nor may "OpenSSL" appear in their names without prior written
   *    permission of the OpenSSL Project.
   *
   * 6. Redistributions of any form whatsoever must retain the following
   *    acknowledgment:
   *    "This product includes software developed by the OpenSSL Project
   *    for use in the OpenSSL Toolkit (http://www.openssl.org/)"
   *
   * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
   * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
   * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
   * PURPOSE ARE DISCLAIMED.  IN NO EVENT SHALL THE OpenSSL PROJECT OR
   * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
   * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
   * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
   * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
   * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
   * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
   * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
   * OF THE POSSIBILITY OF SUCH DAMAGE.
   * ====================================================================
   *
   * This product includes cryptographic software written by Eric Young
   * (eay@cryptsoft.com).  This product includes software written by Tim
   * Hudson (tjh@cryptsoft.com). */
  
  #include <openssl/bn.h>
  
  #include <assert.h>
  #include <string.h>
  
  #include <openssl/cpu.h>
  #include <openssl/err.h>
  #include <openssl/mem.h>
  
  #include "internal.h"
  
  
  #if !defined(OPENSSL_NO_ASM) && defined(OPENSSL_X86_64)
  #define OPENSSL_BN_ASM_MONT5
  #define RSAZ_ENABLED
  
  #include "rsaz_exp.h"
  #endif
  
  int BN_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) {
    int i, bits, ret = 0;
    BIGNUM *v, *rr;
  
    if ((p->flags & BN_FLG_CONSTTIME) != 0) {
      /* BN_FLG_CONSTTIME only supported by BN_mod_exp_mont() */
      OPENSSL_PUT_ERROR(BN, ERR_R_SHOULD_NOT_HAVE_BEEN_CALLED);
      return 0;
    }
  
    BN_CTX_start(ctx);
    if (r == a || r == p) {
      rr = BN_CTX_get(ctx);
    } else {
      rr = r;
    }
  
    v = BN_CTX_get(ctx);
    if (rr == NULL || v == NULL) {
      goto err;
    }
  
    if (BN_copy(v, a) == NULL) {
      goto err;
    }
    bits = BN_num_bits(p);
  
    if (BN_is_odd(p)) {
      if (BN_copy(rr, a) == NULL) {
        goto err;
      }
    } else {
      if (!BN_one(rr)) {
        goto err;
      }
    }
  
    for (i = 1; i < bits; i++) {
      if (!BN_sqr(v, v, ctx)) {
        goto err;
      }
      if (BN_is_bit_set(p, i)) {
        if (!BN_mul(rr, rr, v, ctx)) {
          goto err;
        }
      }
    }
  
    if (r != rr && !BN_copy(r, rr)) {
      goto err;
    }
    ret = 1;
  
  err:
    BN_CTX_end(ctx);
    return ret;
  }
  
  /* maximum precomputation table size for *variable* sliding windows */
  #define TABLE_SIZE 32
  
  typedef struct bn_recp_ctx_st {
    BIGNUM N;  /* the divisor */
    BIGNUM Nr; /* the reciprocal */
    int num_bits;
    int shift;
    int flags;
  } BN_RECP_CTX;
  
  static void BN_RECP_CTX_init(BN_RECP_CTX *recp) {
    BN_init(&recp->N);
    BN_init(&recp->Nr);
    recp->num_bits = 0;
    recp->flags = 0;
  }
  
  static void BN_RECP_CTX_free(BN_RECP_CTX *recp) {
    if (recp == NULL) {
      return;
    }
  
    BN_free(&recp->N);
    BN_free(&recp->Nr);
  }
  
  static int BN_RECP_CTX_set(BN_RECP_CTX *recp, const BIGNUM *d, BN_CTX *ctx) {
    if (!BN_copy(&(recp->N), d)) {
      return 0;
    }
    BN_zero(&recp->Nr);
    recp->num_bits = BN_num_bits(d);
    recp->shift = 0;
  
    return 1;
  }
  
  /* len is the expected size of the result We actually calculate with an extra
   * word of precision, so we can do faster division if the remainder is not
   * required.
   * r := 2^len / m */
  static int BN_reciprocal(BIGNUM *r, const BIGNUM *m, int len, BN_CTX *ctx) {
    int ret = -1;
    BIGNUM *t;
  
    BN_CTX_start(ctx);
    t = BN_CTX_get(ctx);
    if (t == NULL) {
      goto err;
    }
  
    if (!BN_set_bit(t, len)) {
      goto err;
    }
  
    if (!BN_div(r, NULL, t, m, ctx)) {
      goto err;
    }
  
    ret = len;
  
  err:
    BN_CTX_end(ctx);
    return ret;
  }
  
  static int BN_div_recp(BIGNUM *dv, BIGNUM *rem, const BIGNUM *m,
                         BN_RECP_CTX *recp, BN_CTX *ctx) {
    int i, j, ret = 0;
    BIGNUM *a, *b, *d, *r;
  
    BN_CTX_start(ctx);
    a = BN_CTX_get(ctx);
    b = BN_CTX_get(ctx);
    if (dv != NULL) {
      d = dv;
    } else {
      d = BN_CTX_get(ctx);
    }
  
    if (rem != NULL) {
      r = rem;
    } else {
      r = BN_CTX_get(ctx);
    }
  
    if (a == NULL || b == NULL || d == NULL || r == NULL) {
      goto err;
    }
  
    if (BN_ucmp(m, &(recp->N)) < 0) {
      BN_zero(d);
      if (!BN_copy(r, m)) {
        return 0;
      }
      BN_CTX_end(ctx);
      return 1;
    }
  
    /* We want the remainder
     * Given input of ABCDEF / ab
     * we need multiply ABCDEF by 3 digests of the reciprocal of ab */
  
    /* i := max(BN_num_bits(m), 2*BN_num_bits(N)) */
    i = BN_num_bits(m);
    j = recp->num_bits << 1;
    if (j > i) {
      i = j;
    }
  
    /* Nr := round(2^i / N) */
    if (i != recp->shift) {
      recp->shift =
          BN_reciprocal(&(recp->Nr), &(recp->N), i,
                        ctx); /* BN_reciprocal returns i, or -1 for an error */
    }
  
    if (recp->shift == -1) {
      goto err;
    }
  
    /* d := |round(round(m / 2^BN_num_bits(N)) * recp->Nr / 2^(i -
     * BN_num_bits(N)))|
     *    = |round(round(m / 2^BN_num_bits(N)) * round(2^i / N) / 2^(i -
     * BN_num_bits(N)))|
     *   <= |(m / 2^BN_num_bits(N)) * (2^i / N) * (2^BN_num_bits(N) / 2^i)|
     *    = |m/N| */
    if (!BN_rshift(a, m, recp->num_bits)) {
      goto err;
    }
    if (!BN_mul(b, a, &(recp->Nr), ctx)) {
      goto err;
    }
    if (!BN_rshift(d, b, i - recp->num_bits)) {
      goto err;
    }
    d->neg = 0;
  
    if (!BN_mul(b, &(recp->N), d, ctx)) {
      goto err;
    }
    if (!BN_usub(r, m, b)) {
      goto err;
    }
    r->neg = 0;
  
    j = 0;
    while (BN_ucmp(r, &(recp->N)) >= 0) {
      if (j++ > 2) {
        OPENSSL_PUT_ERROR(BN, BN_R_BAD_RECIPROCAL);
        goto err;
      }
      if (!BN_usub(r, r, &(recp->N))) {
        goto err;
      }
      if (!BN_add_word(d, 1)) {
        goto err;
      }
    }
  
    r->neg = BN_is_zero(r) ? 0 : m->neg;
    d->neg = m->neg ^ recp->N.neg;
    ret = 1;
  
  err:
    BN_CTX_end(ctx);
    return ret;
  }
  
  static int BN_mod_mul_reciprocal(BIGNUM *r, const BIGNUM *x, const BIGNUM *y,
                                   BN_RECP_CTX *recp, BN_CTX *ctx) {
    int ret = 0;
    BIGNUM *a;
    const BIGNUM *ca;
  
    BN_CTX_start(ctx);
    a = BN_CTX_get(ctx);
    if (a == NULL) {
      goto err;
    }
  
    if (y != NULL) {
      if (x == y) {
        if (!BN_sqr(a, x, ctx)) {
          goto err;
        }
      } else {
        if (!BN_mul(a, x, y, ctx)) {
          goto err;
        }
      }
      ca = a;
    } else {
      ca = x; /* Just do the mod */
    }
  
    ret = BN_div_recp(NULL, r, ca, recp, ctx);
  
  err:
    BN_CTX_end(ctx);
    return ret;
  }
  
  /* BN_window_bits_for_exponent_size -- macro for sliding window mod_exp
   * functions
   *
   * For window size 'w' (w >= 2) and a random 'b' bits exponent, the number of
   * multiplications is a constant plus on average
   *
   *    2^(w-1) + (b-w)/(w+1);
   *
   * here 2^(w-1)  is for precomputing the table (we actually need entries only
   * for windows that have the lowest bit set), and (b-w)/(w+1)  is an
   * approximation for the expected number of w-bit windows, not counting the
   * first one.
   *
   * Thus we should use
   *
   *    w >= 6  if        b > 671
   *     w = 5  if  671 > b > 239
   *     w = 4  if  239 > b >  79
   *     w = 3  if   79 > b >  23
   *    w <= 2  if   23 > b
   *
   * (with draws in between).  Very small exponents are often selected
   * with low Hamming weight, so we use  w = 1  for b <= 23. */
  #define BN_window_bits_for_exponent_size(b) \
  		((b) > 671 ? 6 : \
  		 (b) > 239 ? 5 : \
  		 (b) >  79 ? 4 : \
  		 (b) >  23 ? 3 : 1)
  
  static int mod_exp_recp(BIGNUM *r, const BIGNUM *a, const BIGNUM *p,
                          const BIGNUM *m, BN_CTX *ctx) {
    int i, j, bits, ret = 0, wstart, window;
    int start = 1;
    BIGNUM *aa;
    /* Table of variables obtained from 'ctx' */
    BIGNUM *val[TABLE_SIZE];
    BN_RECP_CTX recp;
  
    if (BN_get_flags(p, BN_FLG_CONSTTIME) != 0) {
      /* BN_FLG_CONSTTIME only supported by BN_mod_exp_mont() */
      OPENSSL_PUT_ERROR(BN, ERR_R_SHOULD_NOT_HAVE_BEEN_CALLED);
      return 0;
    }
  
    bits = BN_num_bits(p);
  
    if (bits == 0) {
      ret = BN_one(r);
      return ret;
    }
  
    BN_CTX_start(ctx);
    aa = BN_CTX_get(ctx);
    val[0] = BN_CTX_get(ctx);
    if (!aa || !val[0]) {
      goto err;
    }
  
    BN_RECP_CTX_init(&recp);
    if (m->neg) {
      /* ignore sign of 'm' */
      if (!BN_copy(aa, m)) {
        goto err;
      }
      aa->neg = 0;
      if (BN_RECP_CTX_set(&recp, aa, ctx) <= 0) {
        goto err;
      }
    } else {
      if (BN_RECP_CTX_set(&recp, m, ctx) <= 0) {
        goto err;
      }
    }
  
    if (!BN_nnmod(val[0], a, m, ctx)) {
      goto err; /* 1 */
    }
    if (BN_is_zero(val[0])) {
      BN_zero(r);
      ret = 1;
      goto err;
    }
  
    window = BN_window_bits_for_exponent_size(bits);
    if (window > 1) {
      if (!BN_mod_mul_reciprocal(aa, val[0], val[0], &recp, ctx)) {
        goto err; /* 2 */
      }
      j = 1 << (window - 1);
      for (i = 1; i < j; i++) {
        if (((val[i] = BN_CTX_get(ctx)) == NULL) ||
            !BN_mod_mul_reciprocal(val[i], val[i - 1], aa, &recp, ctx)) {
          goto err;
        }
      }
    }
  
    start = 1; /* This is used to avoid multiplication etc
                * when there is only the value '1' in the
                * buffer. */
    wstart = bits - 1; /* The top bit of the window */
  
    if (!BN_one(r)) {
      goto err;
    }
  
    for (;;) {
      int wvalue; /* The 'value' of the window */
      int wend; /* The bottom bit of the window */
  
      if (BN_is_bit_set(p, wstart) == 0) {
        if (!start) {
          if (!BN_mod_mul_reciprocal(r, r, r, &recp, ctx)) {
            goto err;
          }
        }
        if (wstart == 0) {
          break;
        }
        wstart--;
        continue;
      }
  
      /* We now have wstart on a 'set' bit, we now need to work out
       * how bit a window to do.  To do this we need to scan
       * forward until the last set bit before the end of the
       * window */
      wvalue = 1;
      wend = 0;
      for (i = 1; i < window; i++) {
        if (wstart - i < 0) {
          break;
        }
        if (BN_is_bit_set(p, wstart - i)) {
          wvalue <<= (i - wend);
          wvalue |= 1;
          wend = i;
        }
      }
  
      /* wend is the size of the current window */
      j = wend + 1;
      /* add the 'bytes above' */
      if (!start) {
        for (i = 0; i < j; i++) {
          if (!BN_mod_mul_reciprocal(r, r, r, &recp, ctx)) {
            goto err;
          }
        }
      }
  
      /* wvalue will be an odd number < 2^window */
      if (!BN_mod_mul_reciprocal(r, r, val[wvalue >> 1], &recp, ctx)) {
        goto err;
      }
  
      /* move the 'window' down further */
      wstart -= wend + 1;
      start = 0;
      if (wstart < 0) {
        break;
      }
    }
    ret = 1;
  
  err:
    BN_CTX_end(ctx);
    BN_RECP_CTX_free(&recp);
    return ret;
  }
  
  int BN_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, const BIGNUM *m,
                 BN_CTX *ctx) {
    /* For even modulus  m = 2^k*m_odd,  it might make sense to compute
     * a^p mod m_odd  and  a^p mod 2^k  separately (with Montgomery
     * exponentiation for the odd part), using appropriate exponent
     * reductions, and combine the results using the CRT.
     *
     * For now, we use Montgomery only if the modulus is odd; otherwise,
     * exponentiation using the reciprocal-based quick remaindering
     * algorithm is used.
     *
     * (Timing obtained with expspeed.c [computations  a^p mod m
     * where  a, p, m  are of the same length: 256, 512, 1024, 2048,
     * 4096, 8192 bits], compared to the running time of the
     * standard algorithm:
     *
     *   BN_mod_exp_mont   33 .. 40 %  [AMD K6-2, Linux, debug configuration]
     *                     55 .. 77 %  [UltraSparc processor, but
     *                                  debug-solaris-sparcv8-gcc conf.]
     *
     *   BN_mod_exp_recp   50 .. 70 %  [AMD K6-2, Linux, debug configuration]
     *                     62 .. 118 % [UltraSparc, debug-solaris-sparcv8-gcc]
     *
     * On the Sparc, BN_mod_exp_recp was faster than BN_mod_exp_mont
     * at 2048 and more bits, but at 512 and 1024 bits, it was
     * slower even than the standard algorithm!
     *
     * "Real" timings [linux-elf, solaris-sparcv9-gcc configurations]
     * should be obtained when the new Montgomery reduction code
     * has been integrated into OpenSSL.) */
  
    if (BN_is_odd(m)) {
      if (a->top == 1 && !a->neg && BN_get_flags(p, BN_FLG_CONSTTIME) == 0) {
        BN_ULONG A = a->d[0];
        return BN_mod_exp_mont_word(r, A, p, m, ctx, NULL);
      }
  
      return BN_mod_exp_mont(r, a, p, m, ctx, NULL);
    }
  
    return mod_exp_recp(r, a, p, m, ctx);
  }
  
  int BN_mod_exp_mont(BIGNUM *rr, const BIGNUM *a, const BIGNUM *p,
                      const BIGNUM *m, BN_CTX *ctx, BN_MONT_CTX *in_mont) {
    int i, j, bits, ret = 0, wstart, window;
    int start = 1;
    BIGNUM *d, *r;
    const BIGNUM *aa;
    /* Table of variables obtained from 'ctx' */
    BIGNUM *val[TABLE_SIZE];
    BN_MONT_CTX *mont = NULL;
  
    if (BN_get_flags(p, BN_FLG_CONSTTIME) != 0) {
      return BN_mod_exp_mont_consttime(rr, a, p, m, ctx, in_mont);
    }
  
    if (!BN_is_odd(m)) {
      OPENSSL_PUT_ERROR(BN, BN_R_CALLED_WITH_EVEN_MODULUS);
      return 0;
    }
    bits = BN_num_bits(p);
    if (bits == 0) {
      ret = BN_one(rr);
      return ret;
    }
  
    BN_CTX_start(ctx);
    d = BN_CTX_get(ctx);
    r = BN_CTX_get(ctx);
    val[0] = BN_CTX_get(ctx);
    if (!d || !r || !val[0]) {
      goto err;
    }
  
    /* If this is not done, things will break in the montgomery part */
  
    if (in_mont != NULL) {
      mont = in_mont;
    } else {
      mont = BN_MONT_CTX_new();
      if (mont == NULL) {
        goto err;
      }
      if (!BN_MONT_CTX_set(mont, m, ctx)) {
        goto err;
      }
    }
  
    if (a->neg || BN_ucmp(a, m) >= 0) {
      if (!BN_nnmod(val[0], a, m, ctx)) {
        goto err;
      }
      aa = val[0];
    } else {
      aa = a;
    }
  
    if (BN_is_zero(aa)) {
      BN_zero(rr);
      ret = 1;
      goto err;
    }
    if (!BN_to_montgomery(val[0], aa, mont, ctx)) {
      goto err; /* 1 */
    }
  
    window = BN_window_bits_for_exponent_size(bits);
    if (window > 1) {
      if (!BN_mod_mul_montgomery(d, val[0], val[0], mont, ctx)) {
        goto err; /* 2 */
      }
      j = 1 << (window - 1);
      for (i = 1; i < j; i++) {
        if (((val[i] = BN_CTX_get(ctx)) == NULL) ||
            !BN_mod_mul_montgomery(val[i], val[i - 1], d, mont, ctx)) {
          goto err;
        }
      }
    }
  
    start = 1; /* This is used to avoid multiplication etc
                * when there is only the value '1' in the
                * buffer. */
    wstart = bits - 1; /* The top bit of the window */
  
    j = m->top; /* borrow j */
    if (m->d[j - 1] & (((BN_ULONG)1) << (BN_BITS2 - 1))) {
      if (bn_wexpand(r, j) == NULL) {
        goto err;
      }
      /* 2^(top*BN_BITS2) - m */
      r->d[0] = (0 - m->d[0]) & BN_MASK2;
      for (i = 1; i < j; i++) {
        r->d[i] = (~m->d[i]) & BN_MASK2;
      }
      r->top = j;
      /* Upper words will be zero if the corresponding words of 'm'
       * were 0xfff[...], so decrement r->top accordingly. */
      bn_correct_top(r);
    } else if (!BN_to_montgomery(r, BN_value_one(), mont, ctx)) {
      goto err;
    }
  
    for (;;) {
      int wvalue; /* The 'value' of the window */
      int wend; /* The bottom bit of the window */
  
      if (BN_is_bit_set(p, wstart) == 0) {
        if (!start && !BN_mod_mul_montgomery(r, r, r, mont, ctx)) {
          goto err;
        }
        if (wstart == 0) {
          break;
        }
        wstart--;
        continue;
      }
  
      /* We now have wstart on a 'set' bit, we now need to work out how bit a
       * window to do.  To do this we need to scan forward until the last set bit
       * before the end of the window */
      wvalue = 1;
      wend = 0;
      for (i = 1; i < window; i++) {
        if (wstart - i < 0) {
          break;
        }
        if (BN_is_bit_set(p, wstart - i)) {
          wvalue <<= (i - wend);
          wvalue |= 1;
          wend = i;
        }
      }
  
      /* wend is the size of the current window */
      j = wend + 1;
      /* add the 'bytes above' */
      if (!start) {
        for (i = 0; i < j; i++) {
          if (!BN_mod_mul_montgomery(r, r, r, mont, ctx)) {
            goto err;
          }
        }
      }
  
      /* wvalue will be an odd number < 2^window */
      if (!BN_mod_mul_montgomery(r, r, val[wvalue >> 1], mont, ctx)) {
        goto err;
      }
  
      /* move the 'window' down further */
      wstart -= wend + 1;
      start = 0;
      if (wstart < 0) {
        break;
      }
    }
  
    if (!BN_from_montgomery(rr, r, mont, ctx)) {
      goto err;
    }
    ret = 1;
  
  err:
    if (in_mont == NULL) {
      BN_MONT_CTX_free(mont);
    }
    BN_CTX_end(ctx);
    return ret;
  }
  
  /* BN_mod_exp_mont_consttime() stores the precomputed powers in a specific
   * layout so that accessing any of these table values shows the same access
   * pattern as far as cache lines are concerned. The following functions are
   * used to transfer a BIGNUM from/to that table. */
  static int copy_to_prebuf(const BIGNUM *b, int top, unsigned char *buf, int idx,
                            int width) {
    size_t i, j;
  
    if (top > b->top) {
      top = b->top; /* this works because 'buf' is explicitly zeroed */
    }
    for (i = 0, j = idx; i < top * sizeof b->d[0]; i++, j += width) {
      buf[j] = ((unsigned char *)b->d)[i];
    }
  
    return 1;
  }
  
  static int copy_from_prebuf(BIGNUM *b, int top, unsigned char *buf, int idx,
                              int width) {
    size_t i, j;
  
    if (bn_wexpand(b, top) == NULL) {
      return 0;
    }
  
    for (i = 0, j = idx; i < top * sizeof b->d[0]; i++, j += width) {
      ((unsigned char *)b->d)[i] = buf[j];
    }
  
    b->top = top;
    bn_correct_top(b);
    return 1;
  }
  
  /* BN_mod_exp_mont_conttime is based on the assumption that the L1 data cache
   * line width of the target processor is at least the following value. */
  #define MOD_EXP_CTIME_MIN_CACHE_LINE_WIDTH (64)
  #define MOD_EXP_CTIME_MIN_CACHE_LINE_MASK \
    (MOD_EXP_CTIME_MIN_CACHE_LINE_WIDTH - 1)
  
  /* Window sizes optimized for fixed window size modular exponentiation
   * algorithm (BN_mod_exp_mont_consttime).
   *
   * To achieve the security goals of BN_mode_exp_mont_consttime, the maximum
   * size of the window must not exceed
   * log_2(MOD_EXP_CTIME_MIN_CACHE_LINE_WIDTH). 
   *
   * Window size thresholds are defined for cache line sizes of 32 and 64, cache
   * line sizes where log_2(32)=5 and log_2(64)=6 respectively. A window size of
   * 7 should only be used on processors that have a 128 byte or greater cache
   * line size. */
  #if MOD_EXP_CTIME_MIN_CACHE_LINE_WIDTH == 64
  
  #define BN_window_bits_for_ctime_exponent_size(b) \
    ((b) > 937 ? 6 : (b) > 306 ? 5 : (b) > 89 ? 4 : (b) > 22 ? 3 : 1)
  #define BN_MAX_WINDOW_BITS_FOR_CTIME_EXPONENT_SIZE (6)
  
  #elif MOD_EXP_CTIME_MIN_CACHE_LINE_WIDTH == 32
  
  #define BN_window_bits_for_ctime_exponent_size(b) \
    ((b) > 306 ? 5 : (b) > 89 ? 4 : (b) > 22 ? 3 : 1)
  #define BN_MAX_WINDOW_BITS_FOR_CTIME_EXPONENT_SIZE (5)
  
  #endif
  
  /* Given a pointer value, compute the next address that is a cache line
   * multiple. */
  #define MOD_EXP_CTIME_ALIGN(x_)          \
    ((unsigned char *)(x_) +               \
     (MOD_EXP_CTIME_MIN_CACHE_LINE_WIDTH - \
      (((size_t)(x_)) & (MOD_EXP_CTIME_MIN_CACHE_LINE_MASK))))
  
  /* This variant of BN_mod_exp_mont() uses fixed windows and the special
   * precomputation memory layout to limit data-dependency to a minimum
   * to protect secret exponents (cf. the hyper-threading timing attacks
   * pointed out by Colin Percival,
   * http://www.daemonology.net/hyperthreading-considered-harmful/)
   */
  int BN_mod_exp_mont_consttime(BIGNUM *rr, const BIGNUM *a, const BIGNUM *p,
                                const BIGNUM *m, BN_CTX *ctx,
                                BN_MONT_CTX *in_mont) {
    int i, bits, ret = 0, window, wvalue;
    int top;
    BN_MONT_CTX *mont = NULL;
  
    int numPowers;
    unsigned char *powerbufFree = NULL;
    int powerbufLen = 0;
    unsigned char *powerbuf = NULL;
    BIGNUM tmp, am;
  
    top = m->top;
  
    if (!(m->d[0] & 1)) {
      OPENSSL_PUT_ERROR(BN, BN_R_CALLED_WITH_EVEN_MODULUS);
      return 0;
    }
    bits = BN_num_bits(p);
    if (bits == 0) {
      ret = BN_one(rr);
      return ret;
    }
  
    BN_CTX_start(ctx);
  
    /* Allocate a montgomery context if it was not supplied by the caller.
     * If this is not done, things will break in the montgomery part. */
    if (in_mont != NULL) {
      mont = in_mont;
    } else {
      mont = BN_MONT_CTX_new();
      if (mont == NULL || !BN_MONT_CTX_set(mont, m, ctx)) {
        goto err;
      }
    }
  
  #ifdef RSAZ_ENABLED
    /* If the size of the operands allow it, perform the optimized
     * RSAZ exponentiation. For further information see
     * crypto/bn/rsaz_exp.c and accompanying assembly modules. */
    if ((16 == a->top) && (16 == p->top) && (BN_num_bits(m) == 1024) &&
        rsaz_avx2_eligible()) {
      if (NULL == bn_wexpand(rr, 16)) {
        goto err;
      }
      RSAZ_1024_mod_exp_avx2(rr->d, a->d, p->d, m->d, mont->RR.d, mont->n0[0]);
      rr->top = 16;
      rr->neg = 0;
      bn_correct_top(rr);
      ret = 1;
      goto err;
    } else if ((8 == a->top) && (8 == p->top) && (BN_num_bits(m) == 512)) {
      if (NULL == bn_wexpand(rr, 8)) {
        goto err;
      }
      RSAZ_512_mod_exp(rr->d, a->d, p->d, m->d, mont->n0[0], mont->RR.d);
      rr->top = 8;
      rr->neg = 0;
      bn_correct_top(rr);
      ret = 1;
      goto err;
    }
  #endif
  
    /* Get the window size to use with size of p. */
    window = BN_window_bits_for_ctime_exponent_size(bits);
  #if defined(OPENSSL_BN_ASM_MONT5)
    if (window >= 5) {
      window = 5; /* ~5% improvement for RSA2048 sign, and even for RSA4096 */
      if ((top & 7) == 0) {
        powerbufLen += 2 * top * sizeof(m->d[0]);
      }
    }
  #endif
  
    /* Allocate a buffer large enough to hold all of the pre-computed
     * powers of am, am itself and tmp.
     */
    numPowers = 1 << window;
    powerbufLen +=
        sizeof(m->d[0]) *
        (top * numPowers + ((2 * top) > numPowers ? (2 * top) : numPowers));
  #ifdef alloca
    if (powerbufLen < 3072) {
      powerbufFree = alloca(powerbufLen + MOD_EXP_CTIME_MIN_CACHE_LINE_WIDTH);
    } else
  #endif
    {
      if ((powerbufFree = (unsigned char *)OPENSSL_malloc(
              powerbufLen + MOD_EXP_CTIME_MIN_CACHE_LINE_WIDTH)) == NULL) {
        goto err;
      }
    }
  
    powerbuf = MOD_EXP_CTIME_ALIGN(powerbufFree);
    memset(powerbuf, 0, powerbufLen);
  
  #ifdef alloca
    if (powerbufLen < 3072) {
      powerbufFree = NULL;
    }
  #endif
  
    /* lay down tmp and am right after powers table */
    tmp.d = (BN_ULONG *)(powerbuf + sizeof(m->d[0]) * top * numPowers);
    am.d = tmp.d + top;
    tmp.top = am.top = 0;
    tmp.dmax = am.dmax = top;
    tmp.neg = am.neg = 0;
    tmp.flags = am.flags = BN_FLG_STATIC_DATA;
  
  /* prepare a^0 in Montgomery domain */
  /* by Shay Gueron's suggestion */
    if (m->d[top - 1] & (((BN_ULONG)1) << (BN_BITS2 - 1))) {
      /* 2^(top*BN_BITS2) - m */
      tmp.d[0] = (0 - m->d[0]) & BN_MASK2;
      for (i = 1; i < top; i++) {
        tmp.d[i] = (~m->d[i]) & BN_MASK2;
      }
      tmp.top = top;
    } else if (!BN_to_montgomery(&tmp, BN_value_one(), mont, ctx)) {
      goto err;
    }
  
    /* prepare a^1 in Montgomery domain */
    if (a->neg || BN_ucmp(a, m) >= 0) {
      if (!BN_mod(&am, a, m, ctx) ||
          !BN_to_montgomery(&am, &am, mont, ctx)) {
        goto err;
      }
    } else if (!BN_to_montgomery(&am, a, mont, ctx)) {
      goto err;
    }
  
  #if defined(OPENSSL_BN_ASM_MONT5)
    /* This optimization uses ideas from http://eprint.iacr.org/2011/239,
     * specifically optimization of cache-timing attack countermeasures
     * and pre-computation optimization. */
  
    /* Dedicated window==4 case improves 512-bit RSA sign by ~15%, but as
     * 512-bit RSA is hardly relevant, we omit it to spare size... */
    if (window == 5 && top > 1) {
      void bn_mul_mont_gather5(BN_ULONG * rp, const BN_ULONG * ap,
                               const void * table, const BN_ULONG * np,
                               const BN_ULONG * n0, int num, int power);
      void bn_scatter5(const BN_ULONG * inp, size_t num, void * table,
                       size_t power);
      void bn_gather5(BN_ULONG * out, size_t num, void * table, size_t power);
      void bn_power5(BN_ULONG * rp, const BN_ULONG * ap, const void * table,
                     const BN_ULONG * np, const BN_ULONG * n0, int num,
                     int power);
      int bn_from_montgomery(BN_ULONG * rp, const BN_ULONG * ap,
                             const BN_ULONG * not_used, const BN_ULONG * np,
                             const BN_ULONG * n0, int num);
  
      BN_ULONG *np = mont->N.d, *n0 = mont->n0, *np2;
  
      /* BN_to_montgomery can contaminate words above .top
       * [in BN_DEBUG[_DEBUG] build]... */
      for (i = am.top; i < top; i++) {
        am.d[i] = 0;
      }
      for (i = tmp.top; i < top; i++) {
        tmp.d[i] = 0;
      }
  
      if (top & 7) {
        np2 = np;
      } else {
        for (np2 = am.d + top, i = 0; i < top; i++) {
          np2[2 * i] = np[i];
        }
      }
  
      bn_scatter5(tmp.d, top, powerbuf, 0);
      bn_scatter5(am.d, am.top, powerbuf, 1);
      bn_mul_mont(tmp.d, am.d, am.d, np, n0, top);
      bn_scatter5(tmp.d, top, powerbuf, 2);
  
      /* same as above, but uses squaring for 1/2 of operations */
      for (i = 4; i < 32; i *= 2) {
        bn_mul_mont(tmp.d, tmp.d, tmp.d, np, n0, top);
        bn_scatter5(tmp.d, top, powerbuf, i);
      }
      for (i = 3; i < 8; i += 2) {
        int j;
        bn_mul_mont_gather5(tmp.d, am.d, powerbuf, np2, n0, top, i - 1);
        bn_scatter5(tmp.d, top, powerbuf, i);
        for (j = 2 * i; j < 32; j *= 2) {
          bn_mul_mont(tmp.d, tmp.d, tmp.d, np, n0, top);
          bn_scatter5(tmp.d, top, powerbuf, j);
        }
      }
      for (; i < 16; i += 2) {
        bn_mul_mont_gather5(tmp.d, am.d, powerbuf, np2, n0, top, i - 1);
        bn_scatter5(tmp.d, top, powerbuf, i);
        bn_mul_mont(tmp.d, tmp.d, tmp.d, np, n0, top);
        bn_scatter5(tmp.d, top, powerbuf, 2 * i);
      }
      for (; i < 32; i += 2) {
        bn_mul_mont_gather5(tmp.d, am.d, powerbuf, np2, n0, top, i - 1);
        bn_scatter5(tmp.d, top, powerbuf, i);
      }
  
      bits--;
      for (wvalue = 0, i = bits % 5; i >= 0; i--, bits--) {
        wvalue = (wvalue << 1) + BN_is_bit_set(p, bits);
      }
      bn_gather5(tmp.d, top, powerbuf, wvalue);
  
      /* At this point |bits| is 4 mod 5 and at least -1. (|bits| is the first bit
       * that has not been read yet.) */
      assert(bits >= -1 && (bits == -1 || bits % 5 == 4));
  
      /* Scan the exponent one window at a time starting from the most
       * significant bits.
       */
      if (top & 7) {
        while (bits >= 0) {
          for (wvalue = 0, i = 0; i < 5; i++, bits--) {
            wvalue = (wvalue << 1) + BN_is_bit_set(p, bits);
          }
  
          bn_mul_mont(tmp.d, tmp.d, tmp.d, np, n0, top);
          bn_mul_mont(tmp.d, tmp.d, tmp.d, np, n0, top);
          bn_mul_mont(tmp.d, tmp.d, tmp.d, np, n0, top);
          bn_mul_mont(tmp.d, tmp.d, tmp.d, np, n0, top);
          bn_mul_mont(tmp.d, tmp.d, tmp.d, np, n0, top);
          bn_mul_mont_gather5(tmp.d, tmp.d, powerbuf, np, n0, top, wvalue);
        }
      } else {
        const uint8_t *p_bytes = (const uint8_t *)p->d;
        int max_bits = p->top * BN_BITS2;
        assert(bits < max_bits);
        /* |p = 0| has been handled as a special case, so |max_bits| is at least
         * one word. */
        assert(max_bits >= 64);
  
        /* If the first bit to be read lands in the last byte, unroll the first
         * iteration to avoid reading past the bounds of |p->d|. (After the first
         * iteration, we are guaranteed to be past the last byte.) Note |bits|
         * here is the top bit, inclusive. */
        if (bits - 4 >= max_bits - 8) {
          /* Read five bits from |bits-4| through |bits|, inclusive. */
          wvalue = p_bytes[p->top * BN_BYTES - 1];
          wvalue >>= (bits - 4) & 7;
          wvalue &= 0x1f;
          bits -= 5;
          bn_power5(tmp.d, tmp.d, powerbuf, np2, n0, top, wvalue);
        }
        while (bits >= 0) {
          /* Read five bits from |bits-4| through |bits|, inclusive. */
          int first_bit = bits - 4;
          wvalue = *(const uint16_t *) (p_bytes + (first_bit >> 3));
          wvalue >>= first_bit & 7;
          wvalue &= 0x1f;
          bits -= 5;
          bn_power5(tmp.d, tmp.d, powerbuf, np2, n0, top, wvalue);
        }
      }
  
      ret = bn_from_montgomery(tmp.d, tmp.d, NULL, np2, n0, top);
      tmp.top = top;
      bn_correct_top(&tmp);
      if (ret) {
        if (!BN_copy(rr, &tmp)) {
          ret = 0;
        }
        goto err; /* non-zero ret means it's not error */
      }
    } else
  #endif
    {
      if (!copy_to_prebuf(&tmp, top, powerbuf, 0, numPowers) ||
          !copy_to_prebuf(&am, top, powerbuf, 1, numPowers)) {
        goto err;
      }
  
      /* If the window size is greater than 1, then calculate
       * val[i=2..2^winsize-1]. Powers are computed as a*a^(i-1)
       * (even powers could instead be computed as (a^(i/2))^2
       * to use the slight performance advantage of sqr over mul).
       */
      if (window > 1) {
        if (!BN_mod_mul_montgomery(&tmp, &am, &am, mont, ctx) ||
            !copy_to_prebuf(&tmp, top, powerbuf, 2, numPowers)) {
          goto err;
        }
        for (i = 3; i < numPowers; i++) {
          /* Calculate a^i = a^(i-1) * a */
          if (!BN_mod_mul_montgomery(&tmp, &am, &tmp, mont, ctx) ||
              !copy_to_prebuf(&tmp, top, powerbuf, i, numPowers)) {
            goto err;
          }
        }
      }
  
      bits--;
      for (wvalue = 0, i = bits % window; i >= 0; i--, bits--) {
        wvalue = (wvalue << 1) + BN_is_bit_set(p, bits);
      }
      if (!copy_from_prebuf(&tmp, top, powerbuf, wvalue, numPowers)) {
        goto err;
      }
  
      /* Scan the exponent one window at a time starting from the most
       * significant bits.
       */
      while (bits >= 0) {
        wvalue = 0; /* The 'value' of the window */
  
        /* Scan the window, squaring the result as we go */
        for (i = 0; i < window; i++, bits--) {
          if (!BN_mod_mul_montgomery(&tmp, &tmp, &tmp, mont, ctx)) {
            goto err;
          }
          wvalue = (wvalue << 1) + BN_is_bit_set(p, bits);
        }
  
        /* Fetch the appropriate pre-computed value from the pre-buf */
        if (!copy_from_prebuf(&am, top, powerbuf, wvalue, numPowers)) {
          goto err;
        }
  
        /* Multiply the result into the intermediate result */
        if (!BN_mod_mul_montgomery(&tmp, &tmp, &am, mont, ctx)) {
          goto err;
        }
      }
    }
  
    /* Convert the final result from montgomery to standard format */
    if (!BN_from_montgomery(rr, &tmp, mont, ctx)) {
      goto err;
    }
    ret = 1;
  err:
    if (in_mont == NULL) {
      BN_MONT_CTX_free(mont);
    }
    if (powerbuf != NULL) {
      OPENSSL_cleanse(powerbuf, powerbufLen);
      OPENSSL_free(powerbufFree);
    }
    BN_CTX_end(ctx);
    return (ret);
  }
  
  int BN_mod_exp_mont_word(BIGNUM *rr, BN_ULONG a, const BIGNUM *p,
                           const BIGNUM *m, BN_CTX *ctx, BN_MONT_CTX *in_mont) {
    BN_MONT_CTX *mont = NULL;
    int b, bits, ret = 0;
    int r_is_one;
    BN_ULONG w, next_w;
    BIGNUM *d, *r, *t;
    BIGNUM *swap_tmp;
  #define BN_MOD_MUL_WORD(r, w, m)   \
    (BN_mul_word(r, (w)) &&          \
     (/* BN_ucmp(r, (m)) < 0 ? 1 :*/ \
      (BN_mod(t, r, m, ctx) && (swap_tmp = r, r = t, t = swap_tmp, 1))))
    /* BN_MOD_MUL_WORD is only used with 'w' large, so the BN_ucmp test is
     * probably more overhead than always using BN_mod (which uses BN_copy if a
     * similar test returns true). We can use BN_mod and do not need BN_nnmod
     * because our accumulator is never negative (the result of BN_mod does not
     * depend on the sign of the modulus). */
  #define BN_TO_MONTGOMERY_WORD(r, w, mont) \
    (BN_set_word(r, (w)) && BN_to_montgomery(r, r, (mont), ctx))
  
    if (BN_get_flags(p, BN_FLG_CONSTTIME) != 0) {
      /* BN_FLG_CONSTTIME only supported by BN_mod_exp_mont() */
      OPENSSL_PUT_ERROR(BN, ERR_R_SHOULD_NOT_HAVE_BEEN_CALLED);
      return 0;
    }
  
    if (!BN_is_odd(m)) {
      OPENSSL_PUT_ERROR(BN, BN_R_CALLED_WITH_EVEN_MODULUS);
      return 0;
    }
  
    if (m->top == 1) {
      a %= m->d[0]; /* make sure that 'a' is reduced */
    }
  
    bits = BN_num_bits(p);
    if (bits == 0) {
      /* x**0 mod 1 is still zero. */
      if (BN_is_one(m)) {
        ret = 1;
        BN_zero(rr);
      } else {
        ret = BN_one(rr);
      }
      return ret;
    }
    if (a == 0) {
      BN_zero(rr);
      ret = 1;
      return ret;
    }
  
    BN_CTX_start(ctx);
    d = BN_CTX_get(ctx);
    r = BN_CTX_get(ctx);
    t = BN_CTX_get(ctx);
    if (d == NULL || r == NULL || t == NULL) {
      goto err;
    }
  
    if (in_mont != NULL) {
      mont = in_mont;
    } else {
      mont = BN_MONT_CTX_new();
      if (mont == NULL || !BN_MONT_CTX_set(mont, m, ctx)) {
        goto err;
      }
    }
  
    r_is_one = 1; /* except for Montgomery factor */
  
    /* bits-1 >= 0 */
  
    /* The result is accumulated in the product r*w. */
    w = a; /* bit 'bits-1' of 'p' is always set */
    for (b = bits - 2; b >= 0; b--) {
      /* First, square r*w. */
      next_w = w * w;
      if ((next_w / w) != w) {
        /* overflow */
        if (r_is_one) {
          if (!BN_TO_MONTGOMERY_WORD(r, w, mont)) {
            goto err;
          }
          r_is_one = 0;
        } else {
          if (!BN_MOD_MUL_WORD(r, w, m)) {
            goto err;
          }
        }
        next_w = 1;
      }
  
      w = next_w;
      if (!r_is_one) {
        if (!BN_mod_mul_montgomery(r, r, r, mont, ctx)) {
          goto err;
        }
      }
  
      /* Second, multiply r*w by 'a' if exponent bit is set. */
      if (BN_is_bit_set(p, b)) {
        next_w = w * a;
        if ((next_w / a) != w) {
          /* overflow */
          if (r_is_one) {
            if (!BN_TO_MONTGOMERY_WORD(r, w, mont)) {
              goto err;
            }
            r_is_one = 0;
          } else {
            if (!BN_MOD_MUL_WORD(r, w, m)) {
              goto err;
            }
          }
          next_w = a;
        }
        w = next_w;
      }
    }
  
    /* Finally, set r:=r*w. */
    if (w != 1) {
      if (r_is_one) {
        if (!BN_TO_MONTGOMERY_WORD(r, w, mont)) {
          goto err;
        }
        r_is_one = 0;
      } else {
        if (!BN_MOD_MUL_WORD(r, w, m)) {
          goto err;
        }
      }
    }
  
    if (r_is_one) {
      /* can happen only if a == 1*/
      if (!BN_one(rr)) {
        goto err;
      }
    } else {
      if (!BN_from_montgomery(rr, r, mont, ctx)) {
        goto err;
      }
    }
    ret = 1;
  
  err:
    if (in_mont == NULL) {
      BN_MONT_CTX_free(mont);
    }
    BN_CTX_end(ctx);
    return ret;
  }
  
  #define TABLE_SIZE 32
  
  int BN_mod_exp2_mont(BIGNUM *rr, const BIGNUM *a1, const BIGNUM *p1,
                       const BIGNUM *a2, const BIGNUM *p2, const BIGNUM *m,
                       BN_CTX *ctx, BN_MONT_CTX *in_mont) {
    int i, j, bits, b, bits1, bits2, ret = 0, wpos1, wpos2, window1, window2,
                                     wvalue1, wvalue2;
    int r_is_one = 1;
    BIGNUM *d, *r;
    const BIGNUM *a_mod_m;
    /* Tables of variables obtained from 'ctx' */
    BIGNUM *val1[TABLE_SIZE], *val2[TABLE_SIZE];
    BN_MONT_CTX *mont = NULL;
  
    if (!(m->d[0] & 1)) {
      OPENSSL_PUT_ERROR(BN, BN_R_CALLED_WITH_EVEN_MODULUS);
      return 0;
    }
    bits1 = BN_num_bits(p1);
    bits2 = BN_num_bits(p2);
    if (bits1 == 0 && bits2 == 0) {
      ret = BN_one(rr);
      return ret;
    }
  
    bits = (bits1 > bits2) ? bits1 : bits2;
  
    BN_CTX_start(ctx);
    d = BN_CTX_get(ctx);
    r = BN_CTX_get(ctx);
    val1[0] = BN_CTX_get(ctx);
    val2[0] = BN_CTX_get(ctx);
    if (!d || !r || !val1[0] || !val2[0]) {
      goto err;
    }
  
    if (in_mont != NULL) {
      mont = in_mont;
    } else {
      mont = BN_MONT_CTX_new();
      if (mont == NULL) {
        goto err;
      }
      if (!BN_MONT_CTX_set(mont, m, ctx)) {
        goto err;
      }
    }
  
    window1 = BN_window_bits_for_exponent_size(bits1);
    window2 = BN_window_bits_for_exponent_size(bits2);
  
    /* Build table for a1:   val1[i] := a1^(2*i + 1) mod m  for i = 0 ..
     * 2^(window1-1) */
    if (a1->neg || BN_ucmp(a1, m) >= 0) {
      if (!BN_mod(val1[0], a1, m, ctx)) {
        goto err;
      }
      a_mod_m = val1[0];
    } else {
      a_mod_m = a1;
    }
  
    if (BN_is_zero(a_mod_m)) {
      BN_zero(rr);
      ret = 1;
      goto err;
    }
  
    if (!BN_to_montgomery(val1[0], a_mod_m, mont, ctx)) {
      goto err;
    }
  
    if (window1 > 1) {
      if (!BN_mod_mul_montgomery(d, val1[0], val1[0], mont, ctx)) {
        goto err;
      }
  
      j = 1 << (window1 - 1);
      for (i = 1; i < j; i++) {
        if (((val1[i] = BN_CTX_get(ctx)) == NULL) ||
            !BN_mod_mul_montgomery(val1[i], val1[i - 1], d, mont, ctx)) {
          goto err;
        }
      }
    }
  
    /* Build table for a2:   val2[i] := a2^(2*i + 1) mod m  for i = 0 ..
     * 2^(window2-1) */
    if (a2->neg || BN_ucmp(a2, m) >= 0) {
      if (!BN_mod(val2[0], a2, m, ctx)) {
        goto err;
      }
      a_mod_m = val2[0];
    } else {
      a_mod_m = a2;
    }
  
    if (BN_is_zero(a_mod_m)) {
      BN_zero(rr);
      ret = 1;
      goto err;
    }
  
    if (!BN_to_montgomery(val2[0], a_mod_m, mont, ctx)) {
      goto err;
    }
  
    if (window2 > 1) {
      if (!BN_mod_mul_montgomery(d, val2[0], val2[0], mont, ctx)) {
        goto err;
      }
  
      j = 1 << (window2 - 1);
      for (i = 1; i < j; i++) {
        if (((val2[i] = BN_CTX_get(ctx)) == NULL) ||
            !BN_mod_mul_montgomery(val2[i], val2[i - 1], d, mont, ctx)) {
          goto err;
        }
      }
    }
  
    /* Now compute the power product, using independent windows. */
    r_is_one = 1;
    wvalue1 = 0; /* The 'value' of the first window */
    wvalue2 = 0; /* The 'value' of the second window */
    wpos1 = 0;   /* If wvalue1 > 0, the bottom bit of the first window */
    wpos2 = 0;   /* If wvalue2 > 0, the bottom bit of the second window */
  
    if (!BN_to_montgomery(r, BN_value_one(), mont, ctx)) {
      goto err;
    }
  
    for (b = bits - 1; b >= 0; b--) {
      if (!r_is_one) {
        if (!BN_mod_mul_montgomery(r, r, r, mont, ctx)) {
          goto err;
        }
      }
  
      if (!wvalue1 && BN_is_bit_set(p1, b)) {
        /* consider bits b-window1+1 .. b for this window */
        i = b - window1 + 1;
        /* works for i<0 */
        while (!BN_is_bit_set(p1, i)) {
          i++;
        }
        wpos1 = i;
        wvalue1 = 1;
        for (i = b - 1; i >= wpos1; i--) {
          wvalue1 <<= 1;
          if (BN_is_bit_set(p1, i)) {
            wvalue1++;
          }
        }
      }
  
      if (!wvalue2 && BN_is_bit_set(p2, b)) {
        /* consider bits b-window2+1 .. b for this window */
        i = b - window2 + 1;
        while (!BN_is_bit_set(p2, i)) {
          i++;
        }
        wpos2 = i;
        wvalue2 = 1;
        for (i = b - 1; i >= wpos2; i--) {
          wvalue2 <<= 1;
          if (BN_is_bit_set(p2, i)) {
            wvalue2++;
          }
        }
      }
  
      if (wvalue1 && b == wpos1) {
        /* wvalue1 is odd and < 2^window1 */
        if (!BN_mod_mul_montgomery(r, r, val1[wvalue1 >> 1], mont, ctx)) {
          goto err;
        }
        wvalue1 = 0;
        r_is_one = 0;
      }
  
      if (wvalue2 && b == wpos2) {
        /* wvalue2 is odd and < 2^window2 */
        if (!BN_mod_mul_montgomery(r, r, val2[wvalue2 >> 1], mont, ctx)) {
          goto err;
        }
        wvalue2 = 0;
        r_is_one = 0;
      }
    }
  
    if (!BN_from_montgomery(rr, r, mont, ctx)) {
      goto err;
    }
    ret = 1;
  
  err:
    if (in_mont == NULL) {
      BN_MONT_CTX_free(mont);
    }
    BN_CTX_end(ctx);
    return ret;
  }
  